2∆ The Present.

- Does it make sense to say that the present moment is in some sense an 'infinitely thin moment of time'.

Intro.

The essence of this book is that direct observation really seems to tell us that the present is 'all' that there is. Now make no mistake this is remarkable in itself. I cannot in any way explain how or why all this 'stuff', including ourselves, is here. I also couldn't explain how or why it could not be here.

On top of the mystery of how and why everything exists, we seem to think that perhaps we can use our intellect to deduce that not only does all this stuff exist, but it exists 'over time'. But perhaps it is just as valid if we can use our intellect to realise that in fact all this stuff 'just' exists now. And there is only the constantly changing present? does suggesting that an extra, mysterious, and invisible thing called time also exists - really add anything to our understanding? I saying 'it all appeared in the past' really saying any more than 'it all just appeared'?

If time is constantly flowing, and everything is happening now, then just how long is the ‘present moment’,
and can it really be infinitely thin? if so, that really doesn’t leave us much
time to get everything done;

If we express speeds in terms of distance and Time, such as
‘100 kilometres per hour’ then this generally seems to make sense, although we
know the object, say a car doesn’t have to travel the whole 100 kilometres and
take an hour to do so, we know that we are talking about its particular speed
at one point in a journey.

So we could say for example that precisely 300 seconds into
some particular journey our car was travelling at 100 kilometres per hour.

The problem here is that this moment, ‘300 seconds into the
journey’ is just an ‘instant’. That is we are not talking about how far we
actually travelled in say one second, or between 300 and 301 seconds into the
journey. We are declaring that we knew, or could calculate and express the
speed of the car at one particular moment or infinitely thin slice of Time.

This leaves us thinking we have in some way confirmed that
Time exists, and in some way confirmed that it makes sense to think about infinitely
thin slices of Time. And thus that the present moment may be an infinitely thin
slice of time, and thus that we are all ‘In’ one.

This becomes nonsensical however because Einstein concluded
that there was a maximum speed limit in the universe, given by Einstein's
constant ‘C’ and generally referred to as the speed of light XXX clarify why
key here XXX. So if an infinitely thin moment of Time may be expressed as ‘zero
seconds’ then whatever speed we claim our car is travelling at, 100, 200 or
even 300 kilometres per hour, to work out how far it travels in zero seconds we
just multiply the speed by zero.

So at 100 km per hour the distance covered in zero seconds
is zero km.

But if we express speeds in terms of fractions of the speed
of light then we don’t have to consider over what distance a thing may be moving
to still sensibly talk about its speed.

This may not satisfy your mind at this point because you may
want to say ‘yes but we are still talking about how fast the car was going say
300 seconds into its journey, so still talking about an instant of Time’.

The problem here is that if you try to understand how ‘Time
may be nothing more that a man made notion’, from the point of view that ‘Time
really exists’ then you may never see what I am suggesting is true.

Consider as a parallel trying to convince someone that
‘magic’ did not exist who listened to you from the point of view that ‘magic was
real’.

If you show them how one trick can be performed yet appear
to be magic they may accept the point but still claim you haven’t disproved all
possible types of magic.

What I am trying to say is that if you start with the view
that Time does exist then it might seem nonsensical to think of a world in
which things just move and change wherever there is energy flowing.

But if you start with the view that there may be just
objects moving and changing wherever there is energy present, then the idea
that there is also a thing called Time, and infinitely thin slices of it etc
sound equally implausible.

In other words, I personally cannot explain how or why the
universe could possible exist, but that clearly does not mean that I does not
exist.

With equal conviction however, I personally cannot explain
how the universe could possible vanish so that it did not exist.

So I personally confess, and I'm guessing I'm not alone,
that I cannot explain how the universe either could or could not exist.

Similarly when it comes to Time the idea that there is - a
completely intangible thing called ‘Time’ that permeates absolutely everything
but can never be seen, but connects an ever growing ‘past’ that can be seen but
not changed, to an infinitely expansive ‘future’ that can’t be seen but is
unstoppable – seems equally or in fact more unbelievable that the idea that
thing just move and change if there is energy present.

So the idea of Time can seem just as odd as the idea of no
Time.

So at this point it makes sense to go back and reconsider
what you actually see in the world around you.

Figure 13‑1 If time is dilated in
different ways everywhere there is gravity or motion, then 'the present' must
be an extremely distorted, infinitely thin, '3D' slice through '4D' space
time... or is it?

Ultimately all this abstract reasoning may not convince you,
so it can be a good idea to actually go outside, walk around and take a fresh
look at the world around you and confirm what you actually see.

Does everything just exist and move as it appears
to do so right in front of your own eyes, or do all of these moving things
‘require Time’ in which to carry out their motion?

If so do you actually see this Time go by? Or do you just
see objects, including yourself just constantly moving and changing.

When you are looking at a clock hand going round is clock
face really divided into an infinite number of infinitely small segments that
the clock hand passes through infinitely quickly but one by one? And do an
infinite number of slices of zero thickness add up to ‘one’ revolution? Or is
that whole idea abstract over-interpretation?

You may not like the idea of things just existing
and just moving because this word seems to sidestep logic and
reasoning, while discussions on Time usually contain a lot of reasoning.

But by ‘just’ here I mean it as in ‘only’ not as in ‘for no
reason’.

There may or may not be a ‘reason’ why things exist and move
and we may or may not be able to know it, but adding the concept of Time to the
mix doesn’t prove or solve anything at all because without Time the question
is…

‘How can things
just exist and move’?

But/and with Time the question is…

‘How can things
exist and move, in an infinitely thin moment of ‘Time’,

and be connected to and sandwiched
between a massive and ever growing kind of visible but not really ‘thing’ which
apparently can’t even be called a thing but is ‘the past’,

And, be
connected to a probably infinitely large, and completely invisible, though
partially predictable ‘thing’ that isn't a thing, that also can’t ever be
directly seen, called ‘the future’?

So the second version of the question, which includes the
notion of Time, doesn’t answer or explain the first, it is just the first
question again with a lot of extra unexplained baggage.

So is it day time? or night Time? Or is it just light or
dark depending on whether the part of the globe you are on is facing the sun or
not? Do things just move if they get a kick in the right place, or do
they do so over an infinite number of infinitely thin slices of Time?

Is every clock you see fundamentally different from a
rotating playground ride, or a spinning bus wheel?

Does every clock you see ‘measure Time’? Or is it just
another moving object but one specially designed to copy the speed at which the
Earth rotates because we are all stuck on the spinning Earth and knowing
whether we are heading to the sunny side or the dark side is very useful to know?

As with the illusion of vision after too much thinking it is
a good idea to actually go out for a walk and confirm what you actually
perceive, it may be simpler than you think.

In the previous discussion I touched upon the idea of an
infinitely thin slice of Time.

Having read a lot of books on the Time this is a notion that
comes up quite often and is seen as one of the mysteries of Time, but I believe
that the idea of the present moment being an infinitely thin slice of Time in
which, even more incredibly everything happens, can be shown to be the result
of incomplete analysis and misunderstanding of some of the more basic elements
of mathematics and logic.

Does the idea of an hour, minute or second hand constantly
marching through an infinite number of infinitely thin slices actually make sense?

Mathematics is an incredibly useful tool, it allows us to
expand and express our knowledge in incredible ways but problems arise if we
misuse it.

Consider some basic maths. We have a sheep pen in a field
with 6 sheep in it.

If we take 4 sheep out of the pen then 6 minus 4 leaves 2
and we have 2 sheep left. If we take out these 2 sheep then we say 2 minus 2
equals zero, so we have ‘no’ sheep. And a quick inspection of the pen will
confirm that the number of sheep we have is zero.

But to say there are zero sheep in the pen isn't
quite right because it gives zero the status of a number and suggests that
‘zero sheep’ or ‘no sheep’ can in some way exist.

In other words would someone else turning up at the scene
say the same thing? Would they look into an empty pen and declare that what
they saw was ‘zero sheep’ there? what, apart from perhaps some suspicious
droppings, would make them mention sheep, surely our observer is just as
correct in saying there are ‘zero fridges’ or ‘zero footballs’ in the pen.

The point here is that although the number of sheep in the
pen seems to have gone from 6, to 2 to zero, ‘zero’ is not really a number it
is more accurate to call it a ‘place holder’.

If we start thoughtlessly confusing the ‘map’ of mathematics
with the ‘territory’ of the real world we are exploring and in doing so we also
blindly treat zero as a number and expect things to work out we can be lead
into seeing anomalies and paradoxes which really do not exist, for example...

Normally we can divide a given number of things, say 6
sheep, by some other chosen number say 2 or 3.

If two farmers what to share the sheep then 6 divided by 2
means they should get 2 pens and put 3 sheep in each. If we have 3 farmers they
each get 2 sheep but different numbers of farmers might cause serious problems
for the sheep, if we have 6 sheep and 5 farmers then to be shared out fairly
one of the sheep will have to be killed and then chopped up! This is a shame
but not impossible.

If we have a loaf of bread we can cut it into a number of
slices and the thickness of the slices will determine how many slices we can
get from the loaf. Basically it is fairly obvious that the thinner and thinner
we choose to make each slice the more and more slices we will get from any
particular loaf.

A problem might seem to arise if we try to cut the loaf into
slices ‘zero’ cm thick, we might be tempted to say that 1 (loaf) divided into
pieces or slices of zero thickness would give us an infinite number of slices.

But if we look at the actual actions we have to take to cut
a loaf into slices the process is as follows,

If we were cutting slices 2 cm thick and we choose to start
cutting the loaf from the right hand end we would place the knife just at the
right end of the loaf then move it left 2 cm and make a cut, hey presto a 2 cm
slice is made. From here we just repeat the process, moving the knife 2cm to
the left and making another cut, until we run out of loaf.

So, as said if we make the slices thinner, say 1cm then we
only move the knife to the left 1 cm before making each cut, and so we end up
with more slices before we run out of bread.

If we wanted to make slices of zero thickness though we can
do the same thing. Line up the knife with the right hand end of the loaf, then
move it left zero cm and start cutting.

We would find however that we wouldn’t be cutting the bread
at all, just moving the knife down right next to the end of the loaf. And we
could repeat this action a thousand, or even an infinite number of times and we
would always in fact still have a whole loaf.

If we choose to start cutting from somewhere in the middle
of the loaf a similar situation arises, we make the first cut at any point,
thus splitting the loaf into two. Then we move the knife zero distance away
from the first cut, effectively of course not moving it at all. And we cut
again to no effect. This is to say we do not actually create an infinitely thin
slice. As before we could repeat this action of moving the knife zero distance
and cutting again any number of times, up to ‘infinity’ but nothing would
change, we’d still have a single loaf cut in two.

This is why mathematically any number divided by zero is not
said to be infinity but more accurately declared as ‘undefined’.

So using this simple loaf of bread example the idea of an
hour hand on a clock face being something that adds up a infinite number of
infinitely thin slices of Time to make one whole revolution seems hard to
explain, but there is an extremely powerful and practical branch of mathematics
that uses such a process to produce genuinely useful results.

Another interesting feature of mixing mathematics, ‘time’,
and ‘zeros’, is the way we can seem to make logic and common sense seem to fall
apart, or even be fundamentally wrong if we chase zeros and moments
in too much of a frenzy. Consider for example an LP record spinning at a steady
rate on an ordinary record deck.

At any ‘moment’ speck of dust on the outer edge of the
record will be travelling at a certain speed and in a certain direction, while
a similar speck of dust on the opposite edge of the record will have the same
speed but be travelling in the opposite direction.

If we move both specks of dust closer to the centre of the
record, say half way to the central spindle, then they will both still be
moving in opposite directions but at a slower rate. Now the problem is that if
we keep moving the dust specks closer and closer to the central spindle, they
still keep moving, and in opposite directions, but at a slower and slower rate.
So logically, or mathematically, at the very centre of the rotating record
deck, there must be an ‘infinitely small’ ‘point’ that is rotating at a
constant rate, while its ‘edges’ are not moving at all!

Now how something infinitely small can rotate is beyond me,
but I can see for myself that record decks ‘work’, and so it must be within the
reasoning or mathematics where the problem lies. So when considering things
like this, and perhaps matters like ‘is what exists at the very centre of a
black-hole ‘an impossible singularity’?’ we should bear in mind that nature
tends to make things constantly ‘work out’, and ‘add up’ just fine, even if our
understanding and mathematics suggests nature may ‘become impossible’.

In mathematics one of the most powerful and widely used
tools is the function of ‘differentiation’. At this point I can already sense
non-mathematically minded readers shutting down and the mathematicians smelling
blood, so I will tread carefully.

In its simplest form differentiation allows us to work out the
gradient of a line on a graph at any ‘point’. And by point here we mean a
mathematically infinitely small location.

If you were to plot a graph of y = x * x you would see a
curve that started off with no gradient but got steeper, and steeper and
steeper as you followed it to the right of the graph paper.

The gradient or slope of the curve is in fact never a fixed
value such as 1 in 10 or 1 in 5 but mathematically always constantly changing
and different at every point along its length.

Differentiation however allows us to say precisely what the
‘slope’ or rate of change of the line is at any chosen infinitely small
location.

Integration is a related mathematical tool that allows us to
work out the area underneath a section of the kind of graph described above.

Integration works by considering the chosen area beneath
such a graph to be consisting of an infinite number of infinitely thin strips
or rectangles.

To work out the area of a rectangle you multiply its base by
its height, but in this case whatever the height of each rectangle, its base
would be zero. So if we tried to do this, in reality we would be adding up an
infinite number of rectangles each with zero area, and our answer would be
zero.

However integration provides us with the correct commonsense
answer – by leaving us with an equation that has an error margin within it,
but, numerically this ‘error’ ends up being one number or expression, divided
by ‘the number of strips’ we imagine chopping the area under our graph into.
And so the more ‘strips’ we choose the smaller the error, until we effectively,
mathematically, reach an error of ‘zero’[1].XXX
simplify and shorten description? XXX

The point here is that while differentiation works with the
concept of an infinitely thin point on a line still having a gradient or slope,
and integration works with the concept that an infinite number of infinitely
thin slices can add up to a real value, these are in fact both mathematical
tools, that are in reality used in a three dimensional world by three
dimensional mathematicians with 3D brains and 3D pencils just constantly moving
and changing, and transferring graphite to paper in interesting patterns if
they have the energy to do so.

This may sound facetious but I'm just trying to be clear
about what reality is and what is useful but essentially too
‘academic’ here, because otherwise we may have problems further down the line.

For example although
we can work out logically how things like an ‘area beneath a graph’ can be ‘thought
of’ in terms of ‘infinitely thin slices of area’, or how the motion of an
object can be thought of in terms of infinitely thin slices of Time, and
although these methods can produce extremely accurate and useful answers, does this
really clearly and directly prove the existence of infinitely thin slices of
physical areas or infinitely thin moments of Time?[2]

You may see this more clearly from another point of view,
consider how things might look if you had never thought of the idea of Time existing.

Would you consider a mathematical proof that one
can work out the rate at which a gradient on a graph changed over a
theoretically infinitely small span, or the existence of a mathematical method
that allows you to work out the area beneath a section of a graph by dividing
it into an infinite number of infinitely thin strips and adding them together also
as proof that because these operations can be carried out with equations, the
world itself must follow the mathematical tools and therefore also operate this
way.

In other words does the mathematics describe a way of
understanding the movement and change around us, or does it prove that the
movement and change around us is comprises of an infinite number of infinitely
small steps constantly being joined together perfectly, and that a thing called
Time is therefore possible and therefore probably exists? (XXX REPEATED)

The trap is this, when we ‘Time’ the motion of an object we
are generally comparing its motion to the rotation of the Earth. If something
travels 240 km per day then we are really saying that while the Earth smoothly
and constantly rotates (once) the object smoothly and constantly covers a
measured distance (240 km).

To make our figures smaller and more manageable we take on
the idea of imagining that the Earth’s rotation ‘starts’ at some place, and the
idea that we could slice the Earth’s rotation into 24 sections and then we each
of these fictitious or imaginary constructs ‘hours’.

As with basic mathematics as long as we carry out the same
operation on both sides of the equation everything remains equal, so we divide
‘240’ km by 24 to get 10 km. and we say that 240km per day is the same thing as
24 km per hour.

This is not too bad, the problem arises because
mathematically we are able to handle the concept of logically or metaphysically???
Dividing things up into infinite numbers of infinitely thin pieces and then
adding these infinitely thin pieces back together in what turns out to be a very
useful and sensible way.

But then, we make the massive and unfounded assumption that
because we can do this with numbers, and because dividing the ‘day’ into 24
hours (or 1,440 minutes or 86,400 seconds) ‘worked’ then if we can mathematically
divide a day into an infinite number of slices and add them back together ok,
then this in some way proves Time exists and works the same kind of way our
maths does.

We can take a more practical view of this idea of looking at
moments of Time by considering how much fuel a family car might use on a simple
journey.

A car cruising on a motorway at a sensible speed might burn
a single litre of fuel to cover 10 kilometres, but how much fuel does it burn
in ‘an instant’?

To understand just what 10 kilometres on a single litre of
fuel equates to take a look at a litre bottle of water then try to imagine how
thin it would be if it was heated up and stretched out to be 10 km long !

In other words imagine a straight piece of transparent pipe,
10 kilometres long, sealed at both ends and with a diameter such that the
entire 10 km length only held a single litre of fuel ![3]

Now instead of fuelling up a car for a particular journey we
could rig up a somewhat dangerous and highly impractical system whereby the car
picks up all the fuel it needs as it goes along!

To do this we just need a pipe or trough of fuel running
alongside the motorway and to equip the car with a suitable side mounted fuel
scoop.

Now I'm going to take a risk and not patent my above idea,
but here’s the reason for considering it. If you set up a system where by a car
cruising along a motorway is constantly picking up precisely the right amount
of fuel it needs as it goes along then you can always clearly and directly see
and know just how much fuel the car will need for any given distance. In fact,
you wouldn’t even have to measure the distance your were considering.

Figure 13‑2 If a car very dangerously
scooped up its fuel as it went along, the cross-section of the trough would
show the fuel it needed to travel 'zero distance'.

If you set up the fuel trough to have the right cross
sectional area for the particular car, van, bus or tank etc you were running
then it would obviously be a smaller cross section for more efficient machines,
the car or van, and a larger cross section for the bus or tank. So if you make
a pen mark at any point on the ‘fuel trough’ and then another pen mark any
distance away from the first, the amount of fuel in that part of the trough is automatically,
precisely how much fuel the vehicle needs to cover that distance[4].

What is more significant though is that you need only make
one pen mark ! in other words, if you just cut through the fuel trough at any
point the ‘infinitely thin’ cross sectional area of the fuel tells you how much
fuel the vehicle needs ‘in an instant of time’, or ‘ to cover zero
distance’ !

Now at this point the discussion might start to both make
sense and sound ridiculous, so I refer you back to the notion of Time existing,
and therefore the present moment being an infinite collection of infinitely
thin moments of Time. (And say if it sounds ridiculous… then you started it).

It’s interesting to note here that you could use this system
to express the fuel efficiency of any vehicle, or compare the fuel usage of a
group of vehicles, just in terms of hypothetical‘fuel trough cross sections’.

XXX check notes on ‘3d static engine’ are included.- note
static forces and pressures – but all have a direction – not an arrow of time.

The thing to realise here is that if you consider the cross
sectional area of the fuel at the point it is being scooped up by the vehicle
you can see that this area is constant and unchanging.

If you then imagine how the whole system looks, in ‘an
instant’ you can see that while the cross section of fuel can be in some sense
thought of as an infinitely thin 2 dimensional slice, the rest of the car, its
scoop, fuel pipe to the engine, the engine (and car) itself, including all the
cogs, pistons, and cylinders, and fuel exploding in those cylinders, all still
exist as three dimensional things (whether you are thinking about them
as moving or as being stationary in an instant or not).

And if we could magically freeze all this motion, there would
still be ‘frozen’, forces and pressures, stresses and strains in the system,
and even ‘frozen’ momentum in all the ‘moving’ parts. Thus when the ‘re-lease’
button is pressed the whole thing could carry on as if nothing had happened.

The reason for suggesting this odd contraption here is
because this book is about explaining how the world and the motion in it can be
described without the notion of Time, and thus also about describing how the
notion of Time, in this case infinitely thin slices of Time, can be shown as
seeming nonsensical when its consequences are examined. And also the idea of
stopping all motion – although impossible to actually do – can still be thought
about sensibly in a timeless view.

The trick here is to see that the word ‘instant’ may be
invalid. Similar to how the word ‘vanish ‘in the context of ‘magic’ is
invalid.If we assume magic exists then
we have to explain what vanishing is. But if we see we have no legitimate
reason to assume ‘magic’ then the word vanish is seen to be the problem. We
cannot fully explain how the above set up would be in ‘an instant’. But this is
not because the view of timelessness fails, but because the theory of time is
shown up to demand the existence of concepts or things, that are named – but
never shown to exist. In reality we just see three dimensional objects and
motion, so it only makes sense to imagine how the fuel scoop car set up might
seem if it was running very slowly or completely stopped. Not in a frozen, infinitely
thin, instant of time, but just simply all here now but stopped.

[1] Without going in too far into the details this means
that if we imagine the number of slices we cutting our graph area into as being
say 2, 3 or 4 we will have quite a big error (some number divided by 2, 3 or
4); but, if we choose a bigger number of ‘slices’, say 10,000 the error is much
smaller (some number divided by 10,000). And ultimately if we imagine choosing
‘infinity’ slices then our error becomes ‘some number, divided by infinity’,
which tends to give and error of ‘zero’.

[2] Consider for example, the
very idea that we can mathematically calculate an objects exact speed,
at some precise ‘moment’ in a fall, say exactly 10 seconds after release, tends
to start with the unspoken agreement that we are just seeing the object as
being an’ infinitely small point’ that can ‘be’ at precise locations or
infinitely small places, in space.

[3] If you work out just what the diameter of the pipe
would be it may turn out to be much smaller than you might think. Consider that
a litre of volume, water, fuel or whatever, is 1,000 cubic centimetres, and a
cubic centimetre is quite easy to visualise, it’s about the size of a large
sugar cube (CHECK).

If we divide our 10 km, or 10
thousand metres, by 1,000 we get ’10 metres’, and this tells us that our family
car uses one sugar cubed size amount of fuel per 10 metres.

So the car actually uses 1
cubic millimetre of fuel per 10 millimetres of distance ! therefore
the actual cross sectional area of the ‘pipe’ would be one tenth of a square
millimetre (mm squared?) CHECK FIGURES AND UNITS)

[4] We are assuming for
simplicity that the vehicles speed doesn’t matter here.